direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×Dic13, C78⋊4C4, C26⋊5C12, C6.16D26, C78.16C22, C39⋊11(C2×C4), C13⋊7(C2×C12), (C2×C26).3C6, (C2×C78).2C2, C2.2(C6×D13), (C2×C6).2D13, C22.(C3×D13), C26.12(C2×C6), SmallGroup(312,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C6×Dic13 |
Generators and relations for C6×Dic13
G = < a,b,c | a6=b26=1, c2=b13, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C6×Dic13
►Regular action on 312 points
Generators in S312
(1 121 65 80 39 140)(2 122 66 81 40 141)(3 123 67 82 41 142)(4 124 68 83 42 143)(5 125 69 84 43 144)(6 126 70 85 44 145)(7 127 71 86 45 146)(8 128 72 87 46 147)(9 129 73 88 47 148)(10 130 74 89 48 149)(11 105 75 90 49 150)(12 106 76 91 50 151)(13 107 77 92 51 152)(14 108 78 93 52 153)(15 109 53 94 27 154)(16 110 54 95 28 155)(17 111 55 96 29 156)(18 112 56 97 30 131)(19 113 57 98 31 132)(20 114 58 99 32 133)(21 115 59 100 33 134)(22 116 60 101 34 135)(23 117 61 102 35 136)(24 118 62 103 36 137)(25 119 63 104 37 138)(26 120 64 79 38 139)(157 261 209 248 196 287)(158 262 210 249 197 288)(159 263 211 250 198 289)(160 264 212 251 199 290)(161 265 213 252 200 291)(162 266 214 253 201 292)(163 267 215 254 202 293)(164 268 216 255 203 294)(165 269 217 256 204 295)(166 270 218 257 205 296)(167 271 219 258 206 297)(168 272 220 259 207 298)(169 273 221 260 208 299)(170 274 222 235 183 300)(171 275 223 236 184 301)(172 276 224 237 185 302)(173 277 225 238 186 303)(174 278 226 239 187 304)(175 279 227 240 188 305)(176 280 228 241 189 306)(177 281 229 242 190 307)(178 282 230 243 191 308)(179 283 231 244 192 309)(180 284 232 245 193 310)(181 285 233 246 194 311)(182 286 234 247 195 312)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26)(27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52)(53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78)(79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182)(183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208)(209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234)(235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260)(261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286)(287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312)
(1 170 14 157)(2 169 15 182)(3 168 16 181)(4 167 17 180)(5 166 18 179)(6 165 19 178)(7 164 20 177)(8 163 21 176)(9 162 22 175)(10 161 23 174)(11 160 24 173)(12 159 25 172)(13 158 26 171)(27 195 40 208)(28 194 41 207)(29 193 42 206)(30 192 43 205)(31 191 44 204)(32 190 45 203)(33 189 46 202)(34 188 47 201)(35 187 48 200)(36 186 49 199)(37 185 50 198)(38 184 51 197)(39 183 52 196)(53 234 66 221)(54 233 67 220)(55 232 68 219)(56 231 69 218)(57 230 70 217)(58 229 71 216)(59 228 72 215)(60 227 73 214)(61 226 74 213)(62 225 75 212)(63 224 76 211)(64 223 77 210)(65 222 78 209)(79 236 92 249)(80 235 93 248)(81 260 94 247)(82 259 95 246)(83 258 96 245)(84 257 97 244)(85 256 98 243)(86 255 99 242)(87 254 100 241)(88 253 101 240)(89 252 102 239)(90 251 103 238)(91 250 104 237)(105 264 118 277)(106 263 119 276)(107 262 120 275)(108 261 121 274)(109 286 122 273)(110 285 123 272)(111 284 124 271)(112 283 125 270)(113 282 126 269)(114 281 127 268)(115 280 128 267)(116 279 129 266)(117 278 130 265)(131 309 144 296)(132 308 145 295)(133 307 146 294)(134 306 147 293)(135 305 148 292)(136 304 149 291)(137 303 150 290)(138 302 151 289)(139 301 152 288)(140 300 153 287)(141 299 154 312)(142 298 155 311)(143 297 156 310)
G:=sub<Sym(312)| (1,121,65,80,39,140)(2,122,66,81,40,141)(3,123,67,82,41,142)(4,124,68,83,42,143)(5,125,69,84,43,144)(6,126,70,85,44,145)(7,127,71,86,45,146)(8,128,72,87,46,147)(9,129,73,88,47,148)(10,130,74,89,48,149)(11,105,75,90,49,150)(12,106,76,91,50,151)(13,107,77,92,51,152)(14,108,78,93,52,153)(15,109,53,94,27,154)(16,110,54,95,28,155)(17,111,55,96,29,156)(18,112,56,97,30,131)(19,113,57,98,31,132)(20,114,58,99,32,133)(21,115,59,100,33,134)(22,116,60,101,34,135)(23,117,61,102,35,136)(24,118,62,103,36,137)(25,119,63,104,37,138)(26,120,64,79,38,139)(157,261,209,248,196,287)(158,262,210,249,197,288)(159,263,211,250,198,289)(160,264,212,251,199,290)(161,265,213,252,200,291)(162,266,214,253,201,292)(163,267,215,254,202,293)(164,268,216,255,203,294)(165,269,217,256,204,295)(166,270,218,257,205,296)(167,271,219,258,206,297)(168,272,220,259,207,298)(169,273,221,260,208,299)(170,274,222,235,183,300)(171,275,223,236,184,301)(172,276,224,237,185,302)(173,277,225,238,186,303)(174,278,226,239,187,304)(175,279,227,240,188,305)(176,280,228,241,189,306)(177,281,229,242,190,307)(178,282,230,243,191,308)(179,283,231,244,192,309)(180,284,232,245,193,310)(181,285,233,246,194,311)(182,286,234,247,195,312), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260)(261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286)(287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312), (1,170,14,157)(2,169,15,182)(3,168,16,181)(4,167,17,180)(5,166,18,179)(6,165,19,178)(7,164,20,177)(8,163,21,176)(9,162,22,175)(10,161,23,174)(11,160,24,173)(12,159,25,172)(13,158,26,171)(27,195,40,208)(28,194,41,207)(29,193,42,206)(30,192,43,205)(31,191,44,204)(32,190,45,203)(33,189,46,202)(34,188,47,201)(35,187,48,200)(36,186,49,199)(37,185,50,198)(38,184,51,197)(39,183,52,196)(53,234,66,221)(54,233,67,220)(55,232,68,219)(56,231,69,218)(57,230,70,217)(58,229,71,216)(59,228,72,215)(60,227,73,214)(61,226,74,213)(62,225,75,212)(63,224,76,211)(64,223,77,210)(65,222,78,209)(79,236,92,249)(80,235,93,248)(81,260,94,247)(82,259,95,246)(83,258,96,245)(84,257,97,244)(85,256,98,243)(86,255,99,242)(87,254,100,241)(88,253,101,240)(89,252,102,239)(90,251,103,238)(91,250,104,237)(105,264,118,277)(106,263,119,276)(107,262,120,275)(108,261,121,274)(109,286,122,273)(110,285,123,272)(111,284,124,271)(112,283,125,270)(113,282,126,269)(114,281,127,268)(115,280,128,267)(116,279,129,266)(117,278,130,265)(131,309,144,296)(132,308,145,295)(133,307,146,294)(134,306,147,293)(135,305,148,292)(136,304,149,291)(137,303,150,290)(138,302,151,289)(139,301,152,288)(140,300,153,287)(141,299,154,312)(142,298,155,311)(143,297,156,310)>;
G:=Group( (1,121,65,80,39,140)(2,122,66,81,40,141)(3,123,67,82,41,142)(4,124,68,83,42,143)(5,125,69,84,43,144)(6,126,70,85,44,145)(7,127,71,86,45,146)(8,128,72,87,46,147)(9,129,73,88,47,148)(10,130,74,89,48,149)(11,105,75,90,49,150)(12,106,76,91,50,151)(13,107,77,92,51,152)(14,108,78,93,52,153)(15,109,53,94,27,154)(16,110,54,95,28,155)(17,111,55,96,29,156)(18,112,56,97,30,131)(19,113,57,98,31,132)(20,114,58,99,32,133)(21,115,59,100,33,134)(22,116,60,101,34,135)(23,117,61,102,35,136)(24,118,62,103,36,137)(25,119,63,104,37,138)(26,120,64,79,38,139)(157,261,209,248,196,287)(158,262,210,249,197,288)(159,263,211,250,198,289)(160,264,212,251,199,290)(161,265,213,252,200,291)(162,266,214,253,201,292)(163,267,215,254,202,293)(164,268,216,255,203,294)(165,269,217,256,204,295)(166,270,218,257,205,296)(167,271,219,258,206,297)(168,272,220,259,207,298)(169,273,221,260,208,299)(170,274,222,235,183,300)(171,275,223,236,184,301)(172,276,224,237,185,302)(173,277,225,238,186,303)(174,278,226,239,187,304)(175,279,227,240,188,305)(176,280,228,241,189,306)(177,281,229,242,190,307)(178,282,230,243,191,308)(179,283,231,244,192,309)(180,284,232,245,193,310)(181,285,233,246,194,311)(182,286,234,247,195,312), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260)(261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286)(287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312), (1,170,14,157)(2,169,15,182)(3,168,16,181)(4,167,17,180)(5,166,18,179)(6,165,19,178)(7,164,20,177)(8,163,21,176)(9,162,22,175)(10,161,23,174)(11,160,24,173)(12,159,25,172)(13,158,26,171)(27,195,40,208)(28,194,41,207)(29,193,42,206)(30,192,43,205)(31,191,44,204)(32,190,45,203)(33,189,46,202)(34,188,47,201)(35,187,48,200)(36,186,49,199)(37,185,50,198)(38,184,51,197)(39,183,52,196)(53,234,66,221)(54,233,67,220)(55,232,68,219)(56,231,69,218)(57,230,70,217)(58,229,71,216)(59,228,72,215)(60,227,73,214)(61,226,74,213)(62,225,75,212)(63,224,76,211)(64,223,77,210)(65,222,78,209)(79,236,92,249)(80,235,93,248)(81,260,94,247)(82,259,95,246)(83,258,96,245)(84,257,97,244)(85,256,98,243)(86,255,99,242)(87,254,100,241)(88,253,101,240)(89,252,102,239)(90,251,103,238)(91,250,104,237)(105,264,118,277)(106,263,119,276)(107,262,120,275)(108,261,121,274)(109,286,122,273)(110,285,123,272)(111,284,124,271)(112,283,125,270)(113,282,126,269)(114,281,127,268)(115,280,128,267)(116,279,129,266)(117,278,130,265)(131,309,144,296)(132,308,145,295)(133,307,146,294)(134,306,147,293)(135,305,148,292)(136,304,149,291)(137,303,150,290)(138,302,151,289)(139,301,152,288)(140,300,153,287)(141,299,154,312)(142,298,155,311)(143,297,156,310) );
G=PermutationGroup([[(1,121,65,80,39,140),(2,122,66,81,40,141),(3,123,67,82,41,142),(4,124,68,83,42,143),(5,125,69,84,43,144),(6,126,70,85,44,145),(7,127,71,86,45,146),(8,128,72,87,46,147),(9,129,73,88,47,148),(10,130,74,89,48,149),(11,105,75,90,49,150),(12,106,76,91,50,151),(13,107,77,92,51,152),(14,108,78,93,52,153),(15,109,53,94,27,154),(16,110,54,95,28,155),(17,111,55,96,29,156),(18,112,56,97,30,131),(19,113,57,98,31,132),(20,114,58,99,32,133),(21,115,59,100,33,134),(22,116,60,101,34,135),(23,117,61,102,35,136),(24,118,62,103,36,137),(25,119,63,104,37,138),(26,120,64,79,38,139),(157,261,209,248,196,287),(158,262,210,249,197,288),(159,263,211,250,198,289),(160,264,212,251,199,290),(161,265,213,252,200,291),(162,266,214,253,201,292),(163,267,215,254,202,293),(164,268,216,255,203,294),(165,269,217,256,204,295),(166,270,218,257,205,296),(167,271,219,258,206,297),(168,272,220,259,207,298),(169,273,221,260,208,299),(170,274,222,235,183,300),(171,275,223,236,184,301),(172,276,224,237,185,302),(173,277,225,238,186,303),(174,278,226,239,187,304),(175,279,227,240,188,305),(176,280,228,241,189,306),(177,281,229,242,190,307),(178,282,230,243,191,308),(179,283,231,244,192,309),(180,284,232,245,193,310),(181,285,233,246,194,311),(182,286,234,247,195,312)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26),(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182),(183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208),(209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234),(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260),(261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286),(287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312)], [(1,170,14,157),(2,169,15,182),(3,168,16,181),(4,167,17,180),(5,166,18,179),(6,165,19,178),(7,164,20,177),(8,163,21,176),(9,162,22,175),(10,161,23,174),(11,160,24,173),(12,159,25,172),(13,158,26,171),(27,195,40,208),(28,194,41,207),(29,193,42,206),(30,192,43,205),(31,191,44,204),(32,190,45,203),(33,189,46,202),(34,188,47,201),(35,187,48,200),(36,186,49,199),(37,185,50,198),(38,184,51,197),(39,183,52,196),(53,234,66,221),(54,233,67,220),(55,232,68,219),(56,231,69,218),(57,230,70,217),(58,229,71,216),(59,228,72,215),(60,227,73,214),(61,226,74,213),(62,225,75,212),(63,224,76,211),(64,223,77,210),(65,222,78,209),(79,236,92,249),(80,235,93,248),(81,260,94,247),(82,259,95,246),(83,258,96,245),(84,257,97,244),(85,256,98,243),(86,255,99,242),(87,254,100,241),(88,253,101,240),(89,252,102,239),(90,251,103,238),(91,250,104,237),(105,264,118,277),(106,263,119,276),(107,262,120,275),(108,261,121,274),(109,286,122,273),(110,285,123,272),(111,284,124,271),(112,283,125,270),(113,282,126,269),(114,281,127,268),(115,280,128,267),(116,279,129,266),(117,278,130,265),(131,309,144,296),(132,308,145,295),(133,307,146,294),(134,306,147,293),(135,305,148,292),(136,304,149,291),(137,303,150,290),(138,302,151,289),(139,301,152,288),(140,300,153,287),(141,299,154,312),(142,298,155,311),(143,297,156,310)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 12A | ··· | 12H | 13A | ··· | 13F | 26A | ··· | 26R | 39A | ··· | 39L | 78A | ··· | 78AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D13 | Dic13 | D26 | C3×D13 | C3×Dic13 | C6×D13 |
| kernel | C6×Dic13 | C3×Dic13 | C2×C78 | C2×Dic13 | C78 | Dic13 | C2×C26 | C26 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 6 | 12 | 6 | 12 | 24 | 12 |
Matrix representation of C6×Dic13 ►in GL3(𝔽157) generated by
| 144 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 13 |
| 156 | 0 | 0 |
| 0 | 156 | 1 |
| 0 | 13 | 143 |
| 129 | 0 | 0 |
| 0 | 10 | 85 |
| 0 | 21 | 147 |
G:=sub<GL(3,GF(157))| [144,0,0,0,13,0,0,0,13],[156,0,0,0,156,13,0,1,143],[129,0,0,0,10,21,0,85,147] >;
C6×Dic13 in GAP, Magma, Sage, TeX
C_6\times {\rm Dic}_{13} % in TeX
G:=Group("C6xDic13"); // GroupNames label
G:=SmallGroup(312,30);
// by ID
G=gap.SmallGroup(312,30);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-13,60,7204]);
// Polycyclic
G:=Group<a,b,c|a^6=b^26=1,c^2=b^13,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export